ASSIGNMENT 14 for SECTION 001
This assignment is to be handed in. There are two parts: Part A and Part B.
Part A will be graded for completeness. You will receive full marks only if every question has been completed.
Part B will be graded for correctness, clarity and comprehensiveness. In both parts, you must show your
work.
Please submit Part A and Part B separately, with your name on each part.
Part A
From Calculus: Early Transcendentals:
From section 3.10, complete questions: 2, 4, 24, 26, 28, 32(a), 42(a), 44
From section 11.11, complete questions: 14(a), 16(a), 18(a)
From section 4.1, complete questions: 4, 6, 8, 10, 12, 14, 26, 28, 48, 50, 52, 54, 56, 60, 62, 74, 76, 78
From section 4.2, complete questions: 12, 14, 16
Part B
1. Find the Taylor series for cos x at a = 0. (This is known as the Maclaurin series for cos x.)
2. Using your answer from the previous question, evaluate lim
x→0
1 − cos x2
(1 − cos x)
2
.
3. Let f be continuous, with the horizontal asymptotes
lim f (x) = 0 and
x→−∞
lim f (x) = 0.
x→∞
Suppose f (0) = 1. Why does this imply that f has an absolute maximum?
4. For each of the following functions, find all the local and absolute extrema (if any exist):
4. (a) f (x) = x + sin x
4. (b) f (x) =
x99
1.1x
5. Let f be a function which is differentiable on an interval (a, b), with f (x) = 0 at n ≥ 2 points on (a, b).
Prove that f 0 (x) = 0 at at least n − 1 points on (a, b).